The present invention relates to optimization of coordinates of a rough cutting tool at cutting work completion, which optimization is essential in order to suppress self-excited vibration responsible for a decrease in machining accuracy and breakage of a rotating tool or a milling cutter and to perform a stable finish cutting work, in a so-called shoulder cutting, in which a milling cutter is moved to work a groove and a periphery of a boss while rotating.
In a cutting work with a milling cutter, a milling cutter is in some cases low in stiffness to be responsible for generation of that relative vibration between the milling cutter, which vibration is classified into forced vibration and self-excited vibration. A cutting edge of the milling cutter passes through the work to cause a cutting force to act between the milling cutter and the work and the cutting force generates relative displacement whereby the former forced vibration is generated. At this time, the milling cutter or the work vibrates at a cutting frequency determined by a product of the rotating speed of the milling cutter and the number of cutting edges thereof, and in the case where vibration is great, noise and vibration of a milling machine are generated.
On the other hand, with the latter self-excited vibration, there is generated vibration having a frequency close to a natural frequency of the milling cutter. Such vibration has a feature in that it does not occur just after cutting is begun, but vibration is gradually amplified as cutting proceeds. In this case, a natural frequency of a mechanical system is generally several hundreds Hz in many cases and so noise due to the vibration becomes a relatively high sound.
There is established an approach, in which the self-excited vibration is modeled by the regenerative theory of vibration as exemplified by, for example, Y. Altintas and E. Budak: Analytical Prediction of Stability Lobes in Milling, Annals of the CIRP Vol. 44, No. 1 (1995) pages 357 to 362 and predicted in numerical analysis. In the theory, a matter that a milling cutter is increased in vibration together with cutting is called a regenerative effect. Specifically, the self-excited vibration is a phenomenon that in a single-degree-of-freedom analytic model shown in FIG. 1, when a wave surface, which is formed by a cutting edge of a milling cutter 1 one cycle ago cutting a work 2 while vibrating, is cut by a cutting edge, which passes next time, a cutting area 3 become wave-shaped as shown in FIG. 1 and vibration of the milling cutter, which is generated due to variation of a chip thickness of the work 2, is increased as cut proceeds.
In FIG. 1, a chip thickness h(t) of the work 2 at time t is represented by the equation (1) with the use of a tooth passing period Δt, a displacement Δx(t) of the milling cutter, and a chip thickness at first according to a working condition, that is, a feed rate h0.h(t)=h0−Δx(t−Δt)+Δx(t)  (1)
On the other hand, in the case of the single-degree-of-freedom shown in FIG. 1, an equation of motion of the milling cutter is represented by the equation (2) with the use of a mode mass m, which is a factor to determine a compliance transfer function 41, a spring constant k, a damping ratio c, and an external force F.F=m{umlaut over (x)}+c{dot over (x)}+kx  (2)
Also, the external force F in the above-mentioned equation is a cutting force acting between the milling cutter 1 and the work 2 and can be represented by the equation (3) with the use of a chip thickness h(t), an axial depth (a) of cut (an amount, by which the milling cutter 1 cuts in a direction perpendicular to a plane of the drawing in FIG. 1), and a proportional constant K, that is, a cutting constant K determined by a combination of a tool geometry and a work material.F=aKh(t)  (3)
Accordingly, the equation of motion of the milling cutter 1 is given by the equation (4) from the equation (2) and the equation (3).{umlaut over (m)}x+c{dot over (x)}+kx=aKh(t)  (4)
It is possible to evaluate a transfer function of the system represented by the equation (1) and the equation (4) to calculate a stable axial depth (a) of cut for various tooth passing periods Δt.
The tooth passing period Δt can be converted into a rotational frequency when the number of cutting edges of the milling cutter is known. By beforehand predicting a chatter-free axial depth (a) of cut to form a NC program, correction of the NC program due to generation of self-excited vibration is made unnecessary, thus enabling a remarkable reduction in man-hour. In view of a vibrational degree of freedom in X direction and in Y direction, it is possible to obtain a stability lobes in a short period of time without repeatedly calculating an acceleration, velocity and displacement acting between the milling cutter and the work, which correspond to respective points of time in a time domain.
FIGS. 2a and 2b are views illustrating a state, in which a grooving work is performed by the use of a milling cutter 1. FIG. 2a is a cross sectional view taken along a rotational axis of the milling cutter 1 and FIG. 2b is a top view as viewed in a direction along the rotational axis of the milling cutter 1. In this case, when a cutting area 3 (hatched portion), in which a cutting edge 5 of the milling cutter 1 cuts off the work 2, is projected in a feed direction of the milling cutter 1, a rectangular shape shown in FIG. 4a is resulted. Also, when the cutting area 3 is projected in the direction along the rotational axis of the milling cutter 1, a shape defining a part of a crescent as shown in FIG. 4b is resulted. The milling cutter 1 is analyzed by a model, which uses compliance transfer functions 41 and 42 in x and y directions.
Also, in case of further performing a finishing cut on a side of a groove of the work 2, the milling cutter 1 is moved horizontally relative to the work 2 as shown in the cross sectional view of FIG. 3 and the cutting area 3 on the left and the right of the groove is further subjected to a finishing cut.
On the other hand, FIG. 4 shows an example, in which the milling cutter is used to perform a cutting work on an L-shaped corner portion of the work. Here, in case of performing a finishing cut on a side and a bottom surface of a cut portion, on which a rough cutting is performed, it is general that after a finishing cut, in which the cutting area 31 is removed from the side of the work 2 as shown in FIG. 4a, is first performed, the milling cutter 1 is caused to separate from the side of the work 2 as shown in FIG. 4b and the milling cutter 1 is caused to cut into the bottom surface of the work 2.
In this case, since a non-cut portion 33 is formed in a region, in which the side and the bottom surface of the work 2 intersect each other, as apparent from FIG. 4b, it becomes necessary to use the milling cutter 1 to perform a cutting work on the side and the bottom surface of the work 2 in one stroke in order to restrict the non-cut portion to perform a finishing cut in two regions on the side and the bottom surface of the work 2.
As described above, in a shoulder cutting with the use of a milling cutter, self-excited vibration of the milling cutter is liable to occur at the time of a finishing cut and a finishing cut with high accuracy is difficult since a cutting area is not rectangular-shaped but L-shaped in cross section when the finishing cut is performed.
However, a conventional method of predicting a self-excited vibration can accommodate for only the case where the cutting area 3 projected in the feed direction of the rotating tool as shown in FIG. 2 is rectangular-shaped in cross section, and involves a problem that it is not possible to beforehand predict that condition, in which self-excited vibration is not generated in a finishing cut of a work, in which a cutting area is not rectangular-shaped in cross section, in other words, which has a L-shaped cross section, when a finishing cut is performed after a rough cutting. As a result, when a NC program is formed, man-hour is unlimitedly increased, which causes a bottleneck in a finishing cut in shoulder cutting.
The invention is related to an approach to prediction of self-excited vibration of cut to cope with such problem, in other words, prediction of a cutting starting position (coordinates) of a rough cutting tool so that generation of self-excited vibration is suppressed in a finishing cut and further a cutting work at the time of the finishing cut becomes maximum in efficiency.
Prior to describing specific means for solving the problem, an explanation is given to chatter-free axial depth of cut when an oscillatory type of a rotational axis in a milling cutter is maintained in a stable state.
FIG. 5 is a drawing illustrating the case of a two-degree-of-freedom analytic model, that is, the case where a center of a tool during no vibration and the center of the tool during vibration are disposed two-dimensionally away from each other. According to the publication “Analytical Prediction of Stability Lobes in Milling”, those cutting forces Fx and Fy in X direction and in Y direction, which act on a milling cutter 1, is represented by the equation (5) with the use of displacements Δx, Δy of the milling cutter 1 during vibration in X direction and in Y direction where the X direction is a feed direction of the milling cutter.
                              [                                                                      F                  x                                                                                                      F                  y                                                              ]                =                              1            2                    ⁢                      a            ·            Kt            ·                                          [                                                                                                    a                        xx                                                                                                            a                        xy                                                                                                                                                a                        yx                                                                                                            a                        yy                                                                                            ]                            ⁡                              [                                                                                                    Δ                        ⁢                                                                                                  ⁢                        x                                                                                                                                                Δ                        ⁢                                                                                                  ⁢                        y                                                                                            ]                                                                        (        5        )            
However, axx, axy, ayx, ayy, respectively, in the equation (5) are represented by the following equations (6) to (9);
                              a          xx                =                              ∑                          j              =              0                                      N              -              1                                ⁢                                          ⁢                      -                                          g                ⁡                                  (                                      ϕ                    j                                    )                                            ⁡                              [                                                      sin                    ⁢                                                                                  ⁢                    2                    ⁢                                          ϕ                      j                                                        +                                      Kr                    ⁡                                          (                                              1                        -                                                  cos                          ⁢                                                                                                          ⁢                          2                          ⁢                                                      ϕ                            j                                                                                              )                                                                      ]                                                                        (        6        )                                          a          xy                =                              ∑                          j              =              0                                      N              -              1                                ⁢                                          ⁢                      -                                          g                ⁡                                  (                                      ϕ                    j                                    )                                            ⁡                              [                                                      (                                          1                      +                                              cos                        ⁢                                                                                                  ⁢                        2                        ⁢                                                  ϕ                          j                                                                                      )                                    +                                      Kr                    ⁢                                                                                  ⁢                    sin                    ⁢                                                                                  ⁢                    2                    ⁢                                          ϕ                      j                                                                      ]                                                                        (        7        )                                          a          yx                =                              ∑                          j              =              0                                      N              -              1                                ⁢                                          ⁢                                    g              ⁡                              (                                  ϕ                  j                                )                                      ⁡                          [                                                (                                      1                    -                                          cos                      ⁢                                                                                          ⁢                      2                      ⁢                                              ϕ                        j                                                                              )                                -                                  Kr                  ⁢                                                                          ⁢                  sin                  ⁢                                                                          ⁢                  2                  ⁢                                      ϕ                    j                                                              ]                                                          (        8        )                                                      a            yy                    =                                    ∑                              j                =                0                                            N                -                1                                      ⁢                                                  ⁢                                          g                ⁡                                  (                                      ϕ                    j                                    )                                            ⁡                              [                                                      sin                    ⁢                                                                                  ⁢                    2                    ⁢                                          ϕ                      j                                                        -                                      Kr                    ⁡                                          (                                              1                        +                                                  cos                          ⁢                                                                                                          ⁢                          2                          ⁢                                                      ϕ                            j                                                                                              )                                                                      ]                                                    ;                            (        9        )            
where, j indicates number of a cutting edge of a milling cutter, N indicates the number of cutting edges of the milling cutter, Φj indicates a rotating angle of a j-th cutting edge of the milling cutter, a indicates an axial depth of cut of the milling cutter 1, Kt and Kr indicate cutting constants determined by a tool geometry of the milling cutter and a work material, and axx, axy, ayx, ayy, respectively, indicate cutting force factors in x, xy, yx, y directions and are functions of time.
Also, g(Φj) is given by the equation (10);g(φj)=1←φst<φj<φex g(φj)=0←φj<φst,φex<φj;   (10)
where, as shown in FIG. 5, Φst indicates an entry angle of a cutting edge, Φex indicates an exit angle of a cutting edge, it is meant that the milling cutter 1 and the work 2 are in contact with each other in the range of Φst<Φj<Φex, and it is meant that the milling cutter 1 and the work 2 are not in contact with each other in the range of Φj<Φst and Φj>Φex.
Further, the cutting force factors are represented by a matrix (11) and then the equation (5) is converted into the equation (12).
                              [          A          ]                =                  [                                                                      a                  xx                                                                              a                  xy                                                                                                      a                  yx                                                                              a                  yy                                                              ]                                    (        11        )                                          [                                                                      F                  x                                                                                                      F                  y                                                              ]                =                              1            2                    ⁢                      a            ·            Kt            ·                                          [                A                ]                            ⁡                              [                                                                                                    Δ                        ⁢                                                                                                  ⁢                        x                                                                                                                                                Δ                        ⁢                                                                                                  ⁢                        y                                                                                            ]                                                                        (        12        )            
A matrix [A] is a function of a rotating angle Φj of the milling cutter in the equations (6) to (9), that is, a function of time during rotation. In order to make handling of the equation simple, the equation (13) is used to find a matrix [A0] of a time-invariant cutting force;
                                          [                          A              0                        ]                    =                                    1              T                        ⁢                                          ∫                0                T                            ⁢                                                [                  A                  ]                                ⁢                                                                  ⁢                                  ⅆ                  t                                                                    ;                            (        13        )            
where, T indicates a period, during which a cutting edge of the milling cutter performs cutting, and it is shown that a cutting force is integrated over a cutting period of a cutting edge and averaged by the time. Since this is equal to one obtained by integration with respect to a pitch Φp of a cutting edge of the milling cutter and averaging, the equation (13) is rewritten into the equation (14).
                              [                      A            0                    ]                =                              1                          ϕ              p                                ⁢                                    ∫                              ϕ                st                                            ϕ                ex                                      ⁢                                          [                A                ]                            ⁢                                                          ⁢                              ⅆ                ϕ                                                                        (        14        )            
The cutting edge pitch Φp is represented by the equation (15).
                              ϕ          p                =                              2            ⁢            π                    N                                    (        15        )            
By defining the matrix [A0] of a time-invariant cutting force as the equation (16), the following equations (17) to (20) are obtained since cutting force factors αxx, αxy, αyx, αyy, respectively, in the equation (16) are equal to ones obtained by integrating the equations (6) to (9) with respect to a rotating angle Φj.
                              [                      A            0                    ]                =                              N                          2              ⁢              π                                ⁡                      [                                                                                α                    xx                                                                                        α                    xy                                                                                                                    α                    yx                                                                                        α                    yy                                                                        ]                                              (        16        )                                          α          xx                =                                            1              2                        ⁡                          [                                                cos                  ⁢                                                                          ⁢                  2                  ⁢                  ϕ                                -                                  2                  ⁢                                      K                    r                                    ⁢                  ϕ                                +                                                      K                    r                                    ⁢                  sin                  ⁢                                                                          ⁢                  2                  ⁢                  ϕ                                            ]                                            ϕ            st                                ϕ            ex                                              (        17        )                                          α          xy                =                                            1              2                        ⁡                          [                                                                    -                    sin                                    ⁢                                                                          ⁢                  2                  ⁢                  ϕ                                -                                  2                  ⁢                  ϕ                                +                                                      K                    r                                    ⁢                                                                          ⁢                  cos                  ⁢                                                                          ⁢                  2                  ⁢                  ϕ                                            ]                                            ϕ            st                                ϕ            ex                                              (        18        )                                          α          yx                =                                            1              2                        ⁡                          [                                                                    -                    sin                                    ⁢                                                                          ⁢                  2                  ⁢                  ϕ                                +                                  2                  ⁢                  ϕ                                +                                                      K                    r                                    ⁢                                                                          ⁢                  cos                  ⁢                                                                          ⁢                  2                  ⁢                  ϕ                                            ]                                            ϕ            st                                ϕ            ex                                              (        19        )                                          α          yy                =                                            1              2                        ⁡                          [                                                                    -                    cos                                    ⁢                                                                          ⁢                  2                  ⁢                  ϕ                                -                                  2                  ⁢                                      K                    r                                    ⁢                  ϕ                                -                                                      K                    r                                    ⁢                  sin                  ⁢                                                                          ⁢                  2                  ⁢                  ϕ                                            ]                                            ϕ            st                                ϕ            ex                                              (        20        )            
Subsequently, assuming that a tool compliance transfer function Φ(iω) is given by the equation (21) with the use of tool compliance transfer functions Φxx, Φyy, Φxy, Φyx in x, y, xy, and yx directions, displacements Δx, Δy in x and y directions are given by the equation (22) with the use of the equation (21) and cutting forces Fx and Fy in x and y directions, so that a loop transfer function of a vibration system, which is composed of a milling cutter and a work and in which vibration of a cutting edge one cycle ago is fed back to a chip thickness this time by the principle of the regenerative, is represented by the equation (23).
                              [                      Φ            ⁡                          (              ⅈω              )                                ]                =                  [                                                                                          Φ                    xx                                    ⁡                                      (                    ⅈω                    )                                                                                                                    Φ                    xy                                    ⁡                                      (                    ⅈω                    )                                                                                                                                            Φ                    yx                                    ⁡                                      (                    ⅈω                    )                                                                                                                    Φ                    yy                                    ⁡                                      (                    ⅈω                    )                                                                                ]                                    (        21        )                                          [                                                                      Δ                  ⁢                                                                          ⁢                  x                                                                                                      Δ                  ⁢                                                                          ⁢                  y                                                              ]                =                              [                          Φ              ⁡                              (                ⅈω                )                                      ]                    ⁡                      [                                                                                Fx                    ⁡                                          (                      ⅈω                      )                                                                                                                                        Fy                    ⁡                                          (                      ⅈω                      )                                                                                            ]                                              (        22        )                                          F          ⁡                      (            ⅈω            )                          =                              1            2                    ⁢                      a            ·                                                            Kt                  ⁡                                      (                                          1                      -                                              ⅇ                                                  ⅈω                          ⁢                                                                                                          ⁢                          T                                                                                      )                                                  ⁡                                  [                                      A                    0                                    ]                                            ⁡                              [                                  Φ                  ⁡                                      (                    ⅈω                    )                                                  ]                                      ·                          F              ⁡                              (                ⅈω                )                                                                        (        23        )            
From the loop transfer function, a characteristic equation required for discrimination of stability of the vibration system is given by the following equation (24).
                              det          ⁡                      [                                          [                I                ]                            -                                                1                  2                                ⁢                                  Kt                  ·                  a                  ·                                                                                    (                                                  1                          -                                                      ⅇ                                                                                          -                                ⅈω                                                            ⁢                                                                                                                          ⁢                              T                                                                                                      )                                            ⁡                                              [                                                  A                          0                                                ]                                                              ⁡                                          [                                              Φ                        ⁡                                                  (                          ⅈω                          )                                                                    ]                                                                                            ]                          =        0                            (        24        )            
Here, using the equations (25) and (26), the equation (24) becomes the equation (27).
                              [                      Φ            0                    ]                =                              [                          A              0                        ]                    ⁡                      [                          Φ              ⁡                              (                ⅈω                )                                      ]                                              (        25        )                                Λ        =                              -                          N                              4                ⁢                π                                              ⁢                      a            ·                          Kt              ⁡                              (                                  1                  -                                      ⅇ                                                                  -                        ⅈω                                            ⁢                                                                                          ⁢                      T                                                                      )                                                                        (        26        )                                          det          ⁡                      [                                          [                I                ]                            +                              Λ                ⁡                                  [                                                            Φ                      0                                        ⁡                                          (                      ⅈω                      )                                                        ]                                                      ]                          =        0                            (        27        )            Here, Λ indicates an eigenvalue of a matrix [Φ0 (iω)].
As a condition that a vibration system represented by the loop transfer function in the equation (23) should be made stable, it is required that Λ have a negative real part, so that it is possible to calculate the eigenvalue Λ of the matrix [Φ0 (iω)] to calculate a chatter-free axial depth a of cut for a vibration frequency ω and a tooth passing period T with the use of the equation (26).
By the way, in the conventional method of predicting a self-excited vibration, a cutting area of a work is rectangular-shaped in cross section and so a cutting force matrix in the equation (12) is modeled in numerical expression as in the equations (6) to (9), so that a matrix [A0] of a time-invariant cutting force can be represented in numerical expression as in the equations (17) to (20) by the equation (14).
In case of subjecting a side and a bottom surface of a work after rough cutting to a finishing cut in one stroke, which is an object in the invention, however, a cutting area is not rectangular-shaped but L-shaped in cross section, and therefore, analysis must be done by using different entry angles Φst and different exit angles Φex with respect to the case where the side surface is cut, and the case where the bottom surface is cut, respectively.